Combined wavelet transform and regression technique for secondary current compensation of current transformers

F. Li, Y. Li and R.K. Aggarwal

Abstract: A technique is proposed to effectively and efficiently compenyate for secondary current of current transformers. particularly in the presence of remanent flux. The technique first utilises the wavelet transforni to accurately distinguish the distorted section from the healthy section of a secondary current wavefonn. Secondly the technique uses the I-egression method to recover the distorted section making use of features extracted from the healthy section. essentially to restore the secondary current to a scaled down replica of the primary system current. The proposed technique is demonstrated on a samplu power system, achieving an average compensation error of I .5% under varying degrees of saturation caused by different levels of reinanent flux.

1 Introduction

Protective relays play an increasingly important role in cver- stressed electric power systems. For relays providing inaxinium protections in undesirable conditions such as faults, an accurate knowledge of current signals is essential. A practical way of obtaining this knowledge is to use current transformers (CTs) to step down high currcnts in the primary windings to lower levels in secondary windings both in wavefonn shapes and inagni- tude [I]. CTs are ~ioiinally used to feed information into measuring instruments and protective equipment and tils0 to isolate the equipment from the primary circuits to ensure the safety of equipment and personnel. Ideally. the secondary current of a CT should he of the same Waveshape as its corresponding primary current with ii scaled-down magnitude for correct relay operations. CTs widely used in power system consist of thi-ee main parts, namcly. a core of ferrous material, a priinary winding and a secondary winding. The core is magnetised by current in the primary winding. From a modelling point of view, this magnetising efkcct is represented by a magnetising branch in the equivalent circuit of the CT. The current flowing through the branch is the exciting current of the CT: this is one of the major causes of the measurement error, thereby influennng the performance of a CT [ 2 4

A principal cause oFCT error is due to saturation of the iron core, i.e. inability of the latter to support a large flux swing emanating from a large DC offset under transient conditions. Thc saturation effect in a CT will lead to the secondary current to be distorted under transient condi- tions. Very often the unwanted distortior can be accen- tuated with thc presence of renianent flux in the CT core

IEE. Z W I/% Promding.~ online no. 20020296 Dol. l@. l i l49 / ip -p~d:2 i~ i~2~~9~ Publicxion dac: 27rh May 2002. Paper 6ni recavud 13th February 2Dol aiid in revised Corm ?Xi11 N o r e m k 2001 The authors itre with the Department of Elstrunic Uniievsiry oiBsth. Bath RAZ 7AY. VK

Eleclfical Engineering,

which arises by virtue of the fact that when a circuit breaker interrupts a fault. remanent flux gets trapped in the CT core. which could last almost indefinitely under load conditions. If the impedance of the CT load has a phase angle of 60" lag, the remanent flux would reach about 50% of the peak flux value. When a second fault of the same polarity occurs, the transient flux rises l'rom the already trapped remanent flux. As a result, the effective excursion of flux before saturation ensues will be much less. Hence, saturation will take place earlier and will be much more severe. Tests carried out by a CalladIan company have revealed that there are many conditions under which remanent flux is left in a CT core. The tests with CTs on a 230 kV system have shown that 27% cases had between 60 and 80% of peak flux density remaining in the core when the primary circuit breaker was open [4]. Fig. 1 illustrates the significant effect of remanence on the secondary currents.

I

-5 t

The general experience with conventional static or electromagnetic relays does not indicate that remanence is a serious problem [2], but it can cause serious problems with' digital relays. Distorted secondary currents can threaten dependability and security of relays, especially current differential relays; this influence can have a detrimental effect on system operation. Much effort has been put into the development of effective and efficient compensation methods for CT saturation. Papers [ S , 61 present methods of using auxiliary electric equipment for compensation. The equipment has shown satisfactoly compensation results under normal Steady-state condition, hut its performance has as yet to be validated for fault conditions. In papers [7, 8]_ an artificial neural network technique is discussed. However, it can only partially recover distorted currents, and does not take account of remanent flux trapped in a CT core. Another algorithm based on solving differential equations at discrete time steps has been reported in 191. Here again, the effect of remanent flux is not considered.

In this paper, a technique is proposed to compensate for distorted CT secondary currents caused by CT saturation under transient faults, particularly in the presence of varying degrees of remanent flux in the CT core. The proposed technique first uses wavelet analysis to detect the onset of distortion due to CT saturation, and a regression method is then utilised to recover the distorted section using the samples from the healthy section of the waveform. The test results attained from the sample power system under a variety of different system and fault conditions clearly demonstrate satisfactory compensation achieved by the proposed technique.

2 Power system configuration

The sample power system considered comprises of two sources and a single line shown in Fig. 2 to demonstrate the effectiveness of the proposed compensation technique for CT saturation. GI and G2 are equivalent sources, with Z,, and Zsz as their equivalent impedances, respectively. The details of the power system configuration follow.

CT

Fig. 2 Confrgurafion of the sample power s y e m

2.7 Sources

phase-ground voltage: 300 kV RMS frequency: 60 Hz GI positive sequence impedance: 7 +j60Q GI zero sequence impedance: I4+jl20Q G2 positive sequence impedance: 0.45 +j4.5 R G2 zero sequence impedance: 0.45 +j4.5Q

2.2 Transmission Line

length 161 km R1=0.0306Q/km. x1=0.365Q/km, C1 = 12.03nF/km RO=0.304Q/km, XI= 1.358R/!un, CO=7.52nF/km

498

The fault current reaches its limit when a three-phase fault occurs on the transmission line close to the CT end: which is approximately 5kA RMS (Le. 300kV/60Q). Although an extensive series of studies were performed to evaluate the performance of the compensation techniques, the results presented here restrict to single phase-ground and three phase faults lOkm away from the CT location. The EMTP (electromagnetic transients program) package was used to simulate the faults with a sampling rate of 200 samples per cycle.

2.3 CT characteristics Two models are available within the EMTP that can represent CTs [IO]. One is the saturable transformer component. The magnetising branch is represented intem- ally as a nonlinear inductor, whereas 1.A characteristics can he derived using supporting program SATURATION. The other one is the pseudononlinear reactor component (type 96). To use this model, CT hysteresis characteristics should he evaluated in advance. A support routine. HYSTER- ESIS, exists within the EMTP for calculating the character- istics. Although both models give satisfactory results for most simulation cases [I I], the latter was employed for its ease of inittalising remanent flux in CT core using type-96 inductor. The 4005 (a turns ratio of 80) tap was selected from a 6005 ClO0 CT. Its hysteresis characteristic is shown in Fig. 3.

0.4 t saiuialion Doinl

-0.2

-0.4

(9.52, 0.48)

I 0 5 10 15

currem. A

Fig. 3 Mngneti.sution cww of selected CT fap

2.4 CTburden Under maximum fault current, the secondary current (neglecting the magnetising current) becomes 62.5 A (i.e.S000/80). When selecting the 400:s tap, the ClO0 CT can supply 100*(400/600) = 66.67 V. Therefore the max- imum load on the CT secondary is 1.10 (= 66.7 V/62.5A). Because of the existence of lead reactance and secondary reactance, the reactive component of the CT load is h e d as 0.50. Hence the CT burden used is (O.S+JU.5) Q.

3 Principles of proposed compensation technique

The compensation technique presented comprises two stages; the first involves accurately distinguishing between the healthy part and the distortion caused by CT saturation in the current waveform, and the second entails the employment of a regression method to mitigate the effect of distortion in the current waveform through compensa- tion. In this respect, it is important to note that any distortion in the current signals other than that caused by CT saturation is left intact by the proposed compensation technique, and is simply reproduced as a scaled-down version of the input waveforms into the CT primary.

IEE Proc-Gene?, Tramri Dirtrib., Vol. 149, No. 4. July 2002

3.1 Introduction to wavelet analysis The Fourier transform (FT) is a well-known and powerful tool for signal spectral decomposition. However, it has a major drawback of not being able to indicate when sudden changes take place. A solution to this problem is the timc- dependent Fourier transform, also known as the short-time Fourier transform (STFT). STFT processes data in a small section of a signal at a time. which can be looked upon as an invisible window containing a finite number of sample points being analysed. This window moves along the signal as a continuous process until the whole signal is covered. Once the window size is selected with STFT, it remains fixed throughout the signal processing. This approach does, however, affect the accuracy of the information if the window size i s not appropriate. The wavelet transform (WT) overcomes the fixed-window problem by allowing variable-sized windows to be utilised, and gives detailed frequency as well as time-domain information, Le. precisely when a sudden change occurred [12].

A wavelet is a waveform having limited duration with an average value of zero. A typical M uses several versions of a chosen mother wavelet to give different levels of frequency details. These versions are different in their time durations (known as scales in wavelet). Applying WT to a signal is essentially to reveal the extent of similarities between the selected wavelet and the windowed sections in the signal, ultimately to expose different features of the signal, such as abrupt changes, higher harmonics or the trend of the signal. The following is a simplified procedure of WT implementa- tion:

(i) Select a wavelet. (ii) Select the section at the start of the signal (as if the section is put in a window), and compute a coefficient corresponding to the section. (iii) Shift the wavelet to the right of the signal (as if the window is moving), and repeat step (ii) until the whole signal is covered. The combined effect of steps (ii) and (iia) is technically expressed as ‘perform a one-level wavelet decomposition of a signal’. (vi) Scale (stretch) the wavelet (as if the window size is enlarged), and repeat steps (ii) and (iii). Repeat steps (i) to (iv) for all the desired scales, or, all the desired levels.

The above procedure is illustrated in Fig. 4. A low scale means a compressed wavelet and a small window size. so

Signal

wavelet

signal

WaYelet

a

I---- I - L I

that the resultant coefficients reveal high-frequency features. In contrast; a high scale means a stretched wavelet and a large window size, and therefore the coeixcients reveal low frequency features. A coefficient obtained from steps (iii) to (v) represents how closely correlated the wavelet is compared with the section of the signal under study.

The most commonly used WT method is the discrete wavelet transform (DWlJ with which the number of decomposition levels can he easily adjusted to meet various practical requirements. Fig. 5 is an example of a one-level decomposition implemented by DWT, in which the source signal is a truncated sine wave and hence its first derivative (FD) is discontinuous, The first level approximation and the first level detail decomposed from the source signal are also shown in the same Figure.

detail approximation

Fig. 5 tram/onii

Di~corrtitiuily .shown by f irs1 Iecel d a d of dxre le ~ru~der

The source signal is aligned intentionally to the fist detail in Fig. 5 to demonstrate that the first-level detail precisely detects the inception and completion points of FD discontinuity through large magnitudes of resultant coeff- cients. Ths attribute of the DWT has had important applications in power systems: typical examples are monitoring voltage sags and flickers (131 and studying faulted networks [14]. In the study present, the DWT is used primarily for detecting FD discontinuity, that is, to dctect the inception and completion of the distortion of a waveform. The DB4 mother wavelet has been chosen in this study as it best serves the purpose of distortion detection (due to CT saturation) among other mother wavelets available; this is owing to its ability in detecting low amplitude. short duration, fast decaying and oscillating type of signals, typical of those associated with CT saturation.

3.2 Detection of distorted current waveforms using wavelet transform A fault current flowing through the primary circuit o r a CT installed on a transmission line can be considered as the superimposition of a sine component and an exponential component, given as

i ( t ) = C e - ” ‘ + c q s i n a , = i , , ( t ) + i , , ( f ) (1 )

where i = fault current, I = time, ieXp = exponential compo- nent, irin = sine component, E = decaying coefficient of exponential component, 4, =phase angle at time instant I , C= initial value of exponential component, and C, = amplitude of sine component.

If the CT is not saturated, it produces a secondary current of the same shape but with a scaled down value at each sampling point, and the FD of the current signal should be continuous. The first-level detail obtained from

4Y9

DWT decomposition of the secondary current should he similar to the waveform shown in Fig. 6u.

alternative form:

&,.(t) = C4 sin(2n.fr + 4) (5) where /=power frequency and cb = initial phase angle of sine component.

There are now two more unknown parameters C, and $ to be determined. For simplicity. (2n f t t4 ) is replaced by a single variable $ p It is difficult to have one technique to find all the six parameters. Thus an alternative method is sought

secondary current

a b

Fig. 6 detail (I Secondary current not distorted h Secondary current distorted

Di.storrurl purl of second[iry current retided byJir.si l e d

If the CT is Saturated, the secondaly current will he distorted. The first-level detail from its DWT will be discontinuous, as shown in Fig. 6h. The short bursts that contain large coefficients of the first level detail, explicitly mark the start and end of the current distortion, hence clearly distinguishing the distorted section from the healthy section of the current waveform.

3.3 Regression technique Once the distorted scction is separated from the healthy section of a waveform it is possible to use a regression technique to compensate the distorted section with features extracted from the healthy section. Regression is a technique to determine an approximate function by performing a kds l mean square fit to given data [IS]. In this study the technique is used to find approximate secondary current waveform functions by fitting the sample points extracted from the healthy section.

3.3.7 Approximating the exponential compo- nent: The exponential component e-" in (I) can be approximated by a Taylor polynomial. given by:

for determining & Like (4), variable f in is,"([) is substituted hy the number

of sampling points n. Thus the sine component becomes:

isi,,(n) = C4 sin $,, (6) where 4pl = phase angle of nth sampling point. Convention- ally, if the phase angle or the initial point is determined as a reference, the phase angles of the successive points can be easily obtained. However, in real-time, it is difficult to determine the initial phase angle of a fault current. Therefore another sampling point has to be selected to act as the reference. The reference is chosen as the first tuming point (irrespective of whether it is peak or trough) after the distortion has been deduced, which is shown in Fig. 7. The phase angle of this point is assumed as 742; however, it could be - 4 2 . If the turning point is a trough value, the sign of C, and I in (6) is negative; otherwise, it is positive. nni2 denotes its sampling number in Fig. 7.

prim. current

-,. I.

!

d2-2n(n-n,,*)/*r d 2

phase angle

Fig. 7 P h s e ungle of nth .sumpling poim

It has been ascertained through extensive studies that the third-order Taylor polynomial gives a satisfactory accuracy, and any higher-order terms are of secondary importance. Multiplying (2) with C yields

= C" + C,f + C2f2 + c3ri (3) where C, = C_ C, = --St[, C2 = Ca2/2, C3 = -Ca3/6, C,, CI- C? and C3 are unknown coefficients that need to be determined by the regression method. In a digital relay, data is digitised into discrete values through sampling. The variable f for time can be substituted by the number of sampling points n. Thus (3) becomes

i e X p ( n ) = C" + Cln + C2n' + Cp3 (4)

3.2.2 Approximating the sine component: The sine component Cash$, in ( I ) can be rewritten in an

500

Suppose that the secondary current is sampled at a rate of S, samples per cycle, the sampling interval phase angle is 2n/& The phase angle & at sampling point n can be derived from (7)

(7)

3.3.3 Compensating distorted section with regression method: The superimposing of the coinpo- nents discussed in both Sections 3.3.1 and 3.3.2 aves a formula of the form:

i (n ) = C" + ~ l n + Cp' + in' + C, sin 4n ( 8 ) This formula approximately represents a current waveform containing both healthy and distorted sections. In (X), 4n is computed via (71, n is the sampling number, and C,, C,, C2, Ci and C, are unknown coefficients.

Ir;E P m c - G ~ w r . 7iw" Disfrih.. Vul. 149, N o 4, July ?WZ

..... Using the sampled points in the healthy section (the

by the regression method performing the least mean square

number of points depends on the chosen sampling rate within the digital relay), Co, C,, C,, C, and C4 can be solved

3 fitting, thereby compensating for the distorted secondary current. 2 0

4 Results and discussion -5

Up to 20 cases comprising of single-phase to ground and

inception angles ranged from 0 to 180", and remanent flux i from -75 to 80% of the flux at the saturation Doint

three-phase faults were simulated in this study. The 60

(hereafter symbol $,, denotes the CT core flux ai the saturation point). Detailed information of the simulated faults is listed in Table I. Although the study only considers single-phase-to-ground and three-phase faults, the technique is generic enough to apply to other types of faults, such as phase-phase.

Table 1: Initial conditions of simulated faults

Fault type Inception Remanent flux in CT core angle deg

Phaseground 0 VS %b, -0.366 -75

45 -0.144 -30 0 0

Three-phase 90 0.096 20 180 0.29 60

0.38 80

b,: CT core sattiration flux

Figs. 8-12 demonstrate some typical compensation results using the proposed technique for five three-phase- fault cascs. All cases have the different levels of remanent flux in the core. The fault currents comprising the maximum DC offset with such an inception angle will lead to the severest distortion of the secondary currents under certain

5 9

-5

......... beiore comp. - aflercomp.

40 ;e

0 20

i :

0

Fig. 9

i: . .

20 40 60

time, ms

9 5

2 0 L

-5

loo I after compensated

conditions. However, the Figures clearly show that the DWT accurately detects the distorted sections of the original secondary current waveforms, which are clearly marked with rectangles. The Figures also show the primary, original secondary and the compensated secondary currents.

Fig. 8 shows the compensating result in the case of a remanence of -75% $, in the CT core. The polarity of the remanence is opposite to that of the DC offset, which results in a slight distortion beginning at the third cycle of the fault secondary current, as shown in Fig. 8.

The extent of the distortion becomes gradually more severe as the remanence increases rrom a negative value to a positive value. The case shown in Fig. 9 has zero remanence within the CT core. The distortion of the secondary current starts at the second cycle after the fault occurs, one cycle earlier than the case shown in Fig. 8. In Fig. 10, the remanent flux is 20% $,, the secondary current is also distorted at the beginning of the second cycle, but the

501

..... Compensated sec.

9 5

: o $!

-5

100

z b g 50

0

Fig. 11

......... before compensated .

20 40 60 lime. ms

As Fig. 8 hut for 60%; reiiiaiienf pu .~

i

-5

.........

D i :, : . : i

20 40 60 lime. ms

Fig. 12 A s for Fig. 8 bur 80%; renranentpur

waveshape is deformed more severely. With the case shown in Fig. 1 I , when the remanence increases to 60% $.v, the distortion sets in at a much early stage, i.e. the first cycle after the occurrence of the fault.

Fig. 12 illustrates the distortion associated with 80% remanent flux; this is the maximum flux that is possible from a practical CT. Under this situation the transit flux has a very short effective excursion before saturation ensues. It is apparent from the Figure that the distortion of the secondary current is the severest, and sets in only a few milliseconds after the occurrence of a fault.

The cases presented cover a wide range of fault conditions, including the extreme one which is relatively uncommon. Despite the wide variety, all distorted sections of the secondary currents are satisfactorily detected and compensated for by the proposed technique.

The simulations throughout this study were carried out on a 800MHz Celeron PC with the MATLAB 5.2 platform. A typical processing time to process a complete

502

cycle of secondary current waveform, including transform- ing the waveform using db4, detecting the distorted section from healthy section, and then compensating the waveform with the regression technique, was 10ms. In a more practical hardware platform, the processing speed is dependant on the speed of the embedded microprocessor and sampling rate, and the entire signal processing speed is likely to be reduced even further with a high speed processor and appropriate sampling rate.

The overall distribution of the errors of the simulated 20 cases is shown via histograms in Fig. 13. It can he clearly seen that before the compensation the maximum error reaches as high as 80%. and the average error is more than 50%, while after the compensation, the maximum error is 3.5%, and the average is within 2%. Among all the sampling points after compensation, there are only 10% of the points whose error falls in the range of 2-3.5%. They mostly appear in cases with extreme severe conditions.

I 2000

0 50 0 2 4

before aner error I" %

Fig. 13 Seconrirrry current error before ond ufrer mmpensution

5 Conclusions

A new technique to accurately compensate for CT secondary current distortion due to core saturation, particularly in the presence of remanent flux when the distortion is most severe has been demonstrated. The technique first utilises the wavelet transform to accurately distinguish the distorted section from the healthy section of a current waveform; secondly, the technique uses regression to recover the distorted section caused by saturation essentially to restore it to the requisite scaled down replica of the primary system current.

The effectiveness of the proposed technique has been tested on 20 fault cases on a sample system. These faults represent a wide spectrum of conditions: the inception angle range from 0 to 180°, and the remanent flux in the CT core from -75 to 80% of the flux at the saturation point. Despite the wide variations. the technique is able to recover the distorted secondary current from an average error of 50% and a maximum error of 80% before compensation, to 2 and 3.5%, respectively, after the compensation. thereby

I€€ Proc.-Gmcr. Trrmrm. Di.wr6. Vo!. 149, No. 4. July 2G92

clearly demonstrating the effectiveness and robustness of the proposed technique.

Acknowledgment

The authors would like to thank Nuffield Foundation for financial support of this work.

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